I am finally onto my last post on generalised quadrangles. The topic of this one is translation generalised quadrangles. These are a special case of elation generalised quadrangles outlined in my last post. The main source for this post has been Michel Lavrauw’s Phd thesis which is availiable from Ghent’s PhD theses in finite geometry page.

Recall that an elation generalised quadrangle (EGQ) is a generalised quadrangle with a base point ${x}$ and a group of automorphisms that fixes each line incident with ${x}$ and acts regularly on the set of points not collinear with ${x}$. A translation generalised quadrangle (or TGQ) is an EGQ with an abelian elation group. In this case the elation group is referred to as a translation group. We saw in the EGQ post, that ${\mathsf{W}(3,q)}$, for ${q}$ even, has an abelian elation group and so is an example of a TGQ. It has been proved by Stan Payne and Jef Thas that for a TGQ, the elation group must be elementary abelian and in particular ${s}$ and ${t}$ are powers of the same prime.

The focus of this post is elation generalised quadrangles. These are generalised quadrangles defined with respect to certain automorphisms of the generalised quadrangle. My main sources for this post have been Michel (Celle) Lavrauw’s and Maska Law’s Phd theses which are both availiable from Ghent’s PhD theses in finite geometry page.

Let ${\mathcal{Q}}$ be a generalised quadrangle of order ${(s,t)}$ with a point ${x}$. An elation about ${x}$ is an automorphism of ${\mathcal{Q}}$ that fixes ${x}$, fixes each line incident with ${x}$ and fixes no point not collinear with ${x}$. If there exists a group ${G}$ of order ${s^2t}$ consisting entirely of elations and which acts regularly on the set of points not collinear with ${x}$ then ${\mathcal{Q}}$ is called an elation generalised quadrangle or simply an EGQ. The group ${G}$ is called an elation group and ${x}$ is called the base point.

In this post I wish to discuss flock generalised quadrangles. As mentioned in the first of this series, John has already discussed these a bit in a previous post so my main aim will be to flesh that out and provide more background. I have relied heavily on Maska Law’s PhD thesis which is available from Ghent’s PhD theses in finite geometry page.

Recall that a conic ${\mathcal{C}}$ is the set of zeros of a nondegenerate quadratic form on ${\mathrm{PG}(2,q)}$. Embed ${\mathrm{PG}(2,q)}$ as a hyperplane ${\pi}$ in ${\mathrm{PG}(3,q)}$ and take a point ${P}$ not on ${\pi}$. For each of the ${q+1}$ points of ${\mathcal{C}}$ there is a unique line through such a point and ${P}$. Let ${\mathcal{K}}$ be the set of all ${q^2+q+1}$ points on these ${q+1}$ lines. The set ${\mathcal{K}}$ is called a quadratic cone with vertex ${P}$. Now ${\mathsf{P}\Gamma\mathsf{L}(4,q)}$ acts transitively on the set of pairs ${(P,\pi)}$ of points ${P}$ and hyperplanes ${\pi}$ of ${\mathrm{PG}(3,q)}$ where ${\pi}$ does not contain ${P}$, and the stabiliser of such a pair induces ${\mathrm{GL}(3,q)}$ on ${\pi}$ and so acts transitively on the set of conics contained in ${\pi}$ . Thus all quadratic cones of ${\mathrm{PG}(3,q)}$ are equivalent.

An easy way to construct a quadratic cone is to take the zeros of the degenerate quadratic form ${Q(x)=x_2^2-x_1x_3}$, where ${x=(x_1,x_2,x_3,x_4)}$. Here we take ${P}$ to be ${\langle (0,0,0,1)\rangle}$ and ${\pi=\langle (1,0,0,0),(0,1,0,0),(0,0,1,0)\rangle}$. The zeros of ${Q}$ on ${\pi}$ form the conic ${\{\langle (1,t,t^2,0)\rangle\mid t\in\mathsf{GF}(q)\}\cup \{\langle (0,0,1,0) \rangle\}}$. Note that for any plane ${\pi'}$ of ${\mathsf{PG}(3,q)}$ the set of zeros of ${Q}$ on ${\pi'}$ forms a conic. This is all reminiscent of the classical case of a cone in ${\mathbb{R}^3}$, where the intersections of a plane with the cone are the conic sections and are either a point, a circle, an ellipse, a parabola or a hyperbola.

A flock of a quadratic cone ${\mathcal{K}}$ with vertex ${P}$ is a partition of ${\mathcal{K}\backslash\{P\}}$ into ${q}$ disjoint conics. Each conic is the intersection of ${\mathcal{K}}$ with a plane. Let ${\ell}$ be a line of ${\mathrm{PG}(3,q)}$ which intersects ${\mathcal{K}}$ trivially. Then ${\ell}$ is contained in ${q+1}$ planes, one of which contains ${P}$. Each of the remaining ${q}$ planes containing ${\ell}$ meets each of the lines which make up ${\mathcal{K}}$ and hence meets ${\mathcal{K}}$ in ${q+1}$ points. Such a set of ${q+1}$ points is a conic. Morever, since the intersection of two planes through ${\ell}$ is ${\ell}$, the ${q}$ conics we obtain are all disjoint and so we get a flock. Such a flock is called a linear flock.

In this post I want to continue discussing some of the constructions for nonclassical generalised quadrangles. First I need to introduce some new geometrical notions.

Ovals, hyperovals and ovoids

An oval in the projective plane ${\mathrm{PG}(2,q)}$ is a set of ${q+1}$ points such that no three are collinear. The typical example is a conic, that is, the set of zeros of some nondegenerate quadratic form. Up to equivalence we can take this quadratic form to be ${Q(x)=x_1x_2+x_3^2}$, and so any conic can be mapped to the conic defined by ${Q}$ by a collineation of ${\mathrm{PG}(2,q)}$. Lines of the projective plane are either

1. nondegenerate with respect to the quadratic form, and so contain two totally singular points, or
2. degenerate, and contain precisely one totally singular point ${v}$ and this point is perpendicular to all the remaining points on the line, that is, ${B(v,w)=0}$ for all remaining points on the line, where ${B}$ is the bilinear form associated with ${Q}$.

In fact, it is a theorem of Segre that for ${q}$ odd the only ovals are these conics.

For ${q}$ even, any oval can be extended to a set of ${q+2}$ points such that no three are collinear. Such a set is called a hyperoval. This can be seen in the case of a conic where there is a unique point of ${\mathrm{PG}(2,q)}$ which is perpendicular to all points of the conic. This point is referred to as the nucleus. For the quadratic form ${Q(x)}$ given above this is the point ${\langle (0,0,1)\rangle}$. This provides a construction for further ovals in the ${q}$ even case, for any ${q+1}$ points of a hyperoval is an oval. For ${q>4}$ the setwise stabiliser of this hyperoval fixes the nucleus and so these extra ovals are not conics. In general, given an oval ${O}$, for each point ${P}$ on the oval there is a unique line of ${\mathrm{PG}(2,q)}$ meeting ${O}$ at ${P}$ and this line is called a tangent line. The tangent lines meet in a common point outside the conic, and this is the nucleus.

In fact, there are many hyperovals which do not come from adding the nucleus to a conic, and hence lots of ovals. Of note are the Subiaco ovals (named as they were discovered here in Perth and Subiaco is a local suburb, and a pun on the fact that the main football ground in Perth is Subiaco Oval) and the Adelaide ovals (cricket fans will know of the Adelaide oval, the main cricket ground in South Australia). There is also the Lunelli-Sce hyperoval in ${\mathrm{PG}(2,16)}$, which along with the hyperovals formed by adding the nucleus to a conic when ${q=2}$ or ${4}$, are the only hyperovals for which the setwise stabiliser of the hyperoval is transitive on the set of points of the hyperoval. This was proved by Korchmáros in 1978.

Further details on ovals and hyperovals can be found in this survey and at Bill Cherowitzo’s hyperoval page.

John and I have just uploaded to the arxiv a copy of our recent paper, `Point regular automorphism groups of generalised quadrangles‘. We investigate the regular subgroups of some of the known generalised quadrangles. We demonstrate that the class of groups which can act as a point regular group of automorphisms of a generalised quadrangle is much wilder than previously thought.

A permutation group ${G}$ on a set ${\Omega}$ acts regularly on a set ${\Omega}$ if it acts transitively on ${\Omega}$ and only the identity of ${G}$ fixes an element of ${\Omega}$. Studying regular automorphism groups of projective planes has received much attention over the years. Recently attention has turned to the study of groups acting regularly on generalised quadrangles.

Dina Ghinelli proved in 1992 that a Frobenius group or a group with a nontrivial centre cannot act regularly on the points of a generalised quadrangle of order ${(s,s)}$, where ${s}$ is even. Stefaan De Winter and Koen Thas proved in 2006 that if a finite thick generalised quadrangle admits an abelian group of automorphisms acting regularly on its points, then it is the Payne derivation of a translation generalised quadrangle of even order. Satoshi Yoshiara proved that there are no generalised quadrangles of order ${(s^2 , s)}$ admitting an automorphism group acting regularly on points.

In this post I wish to give a construction of some nonclassical generalised quadrangles, that is, ones other than ${\mathsf{H}(3,q^2),\mathsf{H}(4,q^2), \mathsf{W}(3,q),\mathsf{Q}(4,q),\mathsf{Q}^-(5,q)}$ and the dual of ${\mathsf{H}(4,q^2)}$. These were discussed in the previous post.

The construction I will discuss is known as Payne derivation and is due to Stan Payne. It constructs new GQs from old ones.

Regular points

First I need to introduce some terminology and discuss the notion of a regular point. Let ${\mathcal{Q}}$ be a generalised quadrangle of order ${(s,t)}$. Let ${x,y}$ be points of ${\mathcal{Q}}$. We say that ${x}$ is collinear to ${y}$ if there is a line of the GQ containing both ${x}$ and ${y}$. We usually denote this by ${x\sim y}$ and if ${x}$ and ${y}$ are not collinear we write ${x\not\sim y}$. For a set ${S}$ of points let ${S^{\perp}}$ denote the set of points collinear with each element of ${S}$. We say that a point ${x}$ is incident with a line ${\ell}$ if ${\ell}$ contains ${x}$. Again we denote this by ${x\sim \ell}$.

Now suppose that ${x,y}$ are two points such that ${x}$ is not collinear with ${y}$. For each line ${\ell}$ incident with ${x}$, since ${y}$ is not on ${\ell}$ there is a unique point on ${\ell}$ incident with ${y}$. Since ${x}$ lies on ${t+1}$ lines, it follows that ${\{x,y\}^{\perp}}$ has size ${t+1}$. Moreover, for each ${u,v\in\{x,y\}^{\perp}}$ we have that ${u}$ is not incident with ${v}$, otherwise ${x,u,v}$ would be a triangle in the geometry. Thus ${\{u,v\}^\perp}$ has size ${t+1}$. Hence ${\{x,y\}^{\perp\perp}}$ has size at most ${t+1}$. Note that ${x,y\in\{x,y\}^{\perp\perp}}$ so we in fact have that ${2\leq |\{x,y\}^{\perp\perp}|\leq t+1}$. If ${\{x,y\}^{\perp\perp}=t+1}$ for all points ${y}$ not collinear with ${x}$ we say that ${x}$ is a regular point.

As part of John’s goal to turn me into a geometer I have been doing some research lately into generalised quadrangles.  To help me in understanding all the various ways of constructing them, I have decided to write a series of posts outlining the known generalised quadrangles and in particular looking at what their automorphism groups are.

Regular readers of this blog and especially followers of the posts from our Buildings, geometries and algebraic groups study group,  will be familar with the basic definitions and the classical examples. They were first introduced in the second post of that series and generalised polygons were introduced in the eighth. Actually the term “generalised quadrangle’ is one of the more prominent tags in the tag cloud.

Recapping, a generalised quadrangle is a point-line geometry satisfying the following two axioms:

• Any two points lie on at most one line.
• Given  a line $\ell$ and a point p not incident with $\ell$, there is a unique point q on $\ell$ which is incident with p

The second axiom implies that there are no triangles in the geometry, that is, there is no triple of points with any pair collinear. A generalised quadrangle is said to have order $(s,t)$ if each line contains $s+1$ points and each point lies on $t+1$ lines. If $s=t$ we usually just say that the generalised quadrangle has order $s$.  If $s,t\geq 2$, the quadrangle is said to be thick.

Given a generalised quadrangle $\mathcal{Q}$ of order $(s,t)$ we can define a new generalised quadrangle whose points are the lines of $\mathcal{Q}$, whose lines are the points of $\mathcal{Q}$ and incidence is inherited from $\mathcal{Q}$. This new GQ has order $(t,s)$ and is called the dual of $\mathcal{Q}$.

Recently I’ve been working with generalised quadrangles which arise from flocks, and in particular, I use the model introduced by Norbert Knarr in his 1992 paper “A geometric construction of generalized quadrangles from polar spaces of rank three”. However, I’m not going to tell you everything because it would be very long. I’m not going to tell you what a generalised quadrangle is, what a flock is or even why we ought to care about such objects. Instead, my purpose is to de-mystify the Knarr model by explaining where it comes from for the classical object, the simplest case. So really, this post is intended for someone who is already familiar with the topic, but maybe later I or someone else will trace backwards to where all this began.

The three-dimensional finite Hermitian variety $H(3,q^2)$ can be constructed in the following way. Consider the following Hermitian form on a four-dimensional vector space over $GF(q^2)$:

$\langle x,y\rangle :=x_1y_1^q+x_2y_2^q+x_3y_3^q+x_4y_4^q.$

Let the points be the one-dimensional subspaces for which this form restricts to the zero form, and let the lines be the two-dimensional subspaces for which this form computes only zero on them. Then this geometry of points and lines gives us a generalised quadrangle, and it is the classical example in the category of flock quadrangles. In other words, this geometry is a classical polar space of rank 2. Now the vectors of the vector space $GF(q^2)^4$ are certainly in one-to-one correspondence with the vectors of $GF(q)^8$, but in fact, more can be said. We can define a bilinear form over $GF(q)$ by $B(u,v):=\gamma(\langle u,v\rangle -\langle u,v\rangle^q)$, where $\gamma$ is some element of $GF(q^2)$ such that $\gamma=-\gamma^q$, and we see that is alternating since

$B(v,u) = \gamma(\langle v,u\rangle -\langle v,u\rangle^q)=\gamma(\langle u,v\rangle^q -\langle u,v\rangle)=-B(u,v)$.

So this form defines a symplectic space $W(7,q)$ on $GF(q)^8$. What is interesting in this correspondence is that the points of $H(3,q^2)$ go to lines of $W(7,q)$, and the lines of $H(3,q^2)$ go to solids of $W(7,q)$. Now take a point P of $H(3,q^2)$. Then P maps to a line P’ of $W(7,q)$. We then take an arbitrary point X on this line P’ and note that P’ is contained in $X^\perp$. Now we project to the quotient polar space $X^\perp/X$ (which is isomorphic to $W(5,q)$) via the map $U\mapsto (X^\perp\cap \langle X, U\rangle)/X$. So we obtain a map from totally isotropic subspaces of $H(3,q^2)$ to totally isotropic subspaces of $W(5,q)$, and something interesting happens with respect to the point P that we started with:

 $P$ a point R of W(5,q) lines on $P$ a set of t.i. planes $\pi_i$ on R such that the projection to $R^\perp/R$ yields a BLT-set of lines of W(3,q) lines not on $P$ t.i. planes which meet some $\pi_i$ in a line not on R points collinear with $P$ lines of W(5,q) contained in some $\pi_i$ points not-collinear with $P$ points of W(5,q) not in the perp of R.

This is the Knarr model of a flock generalised quadrangle, where the input is a BLT-set of lines of $W(3,q)$ (not just the one obtained from a pencil of lines of $H(3,q^2)$). So essentially the Knarr model of a flock generalised quadrangle is the generalisation of the field reduction of $H(3,q^2)$, where we change the BLT-set in the resulting geometry of $W(5,q)$.

Here is the next instalment in our Phan systems study group which was held yesterday.  I forgot to bring my laptop  so I took notes by hand. Blogging the study group has already had one benefit as Gordon is now attending. Having been initially put off by the title he has realised that we are actually doing lots of stuff that he is interested in learning.

John began the discussion by working through the geometry of the symplectic polar space of rank 2.  Let $J=\left(\begin{array}{cc} 0_2 &I_2\\ -I_2& 0_2\end{array}\right)$ and let $\beta$ be the bilinear form on $V=K^4$, for some field K, given by $\beta(v,w)=vJw^T$. Note we are using row vectors, and if $v=(v_1,v_2,v_3,v_4)$, $w=(w_1,w_2,w_3,w_4)$ we have $\beta(v,w)=v_1w_3-v_3w_1+v_2w_4-v_4w_2$. Then $\beta$ is an alternating form, that is $\beta(v,w)=-\beta(w,v)$ and $\beta(v,v)=0$ for all $v,w\in V$. The totally isotropic subspaces of V, are those subspaces W for which the restriction $\beta_{\mid W}$ is zero, that is, $\beta(v,w)=0$ for all $v,w\in W$. We can then form a point-line geometry whose points are the totally isotropic 1-spaces of V and the lines are the totally isotropic 2-spaces of V.  Note that all 1-spaces of V are totally isotropic, but V does not contain any totally isotropic 3-spaces. A hyperbolic pair is a pair of vectors $(e,f)$ such that $\beta(e,f)=1$.

Let $e_1=(1,0,0,0), e_2=(0,1,0,0), f_1=(0,0,1,0)$ and $f_2=(0,0,0,1)$. Then $(e_1,f_1)$ and $(e_2,f_2)$ are hyperbolic pairs, while $\langle e_1,e_2\rangle,\langle f_1,f_2\rangle,\langle e_1,f_2\rangle$ and $\langle e_2,f_1\rangle$ are totally isotropic. We have $V=\langle e_1,f_1\rangle\perp\langle e_2,f_2\rangle$, that is, V can be written as an orthogonal direct sum of two hyperbolic lines. One of the chambers in this geometry is $\langle e_1\rangle\subset \langle e_1,e_2\rangle$.

The point-line geometry we have just defined is an example of a generalised quadrangle. That is, it satisfies the following two axioms:

• Any two points lie on at most one line.
• Given  a line $\ell$ and a point p not incident with $\ell$, there is a unique point q on $\ell$ which is incident with p.

Such a geometry is often referred to as a geometry of type $B_2$  and is denoted by the diagram